I've heard about the four square theorem and how it was proven using quaternions, which I found to be extremely fascinating, I was wondering whether there are any other interesting theorems which seem to have nothing to do with quaternions that were proven with quaternions, I looked up online but couldn't find any other than four square theorem, and there must be more. Thanks!
What interesting mathematical identities (or theorems) have been proven using quaternions
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I was impressed by Bernstein and Robinson's 1966 proof that if some polynomial of an operator on a Hilbert space is compact then the operator has an invariant subspace. This solved a particular instance of invariant subspace problem, one of pure operator theory without any hint of logic.
Bernstein and Robinson used hyperfinite-dimensional Hilbert space, a nonstandard model, and some very metamathematical things like transfer principle and saturation. Halmos was very unhappy with their proof and eliminated non-standard analysis from it the same year. But the fact remains that the proof was originally found through non-trivial application of the model theory.
Another example is the solution to the Hilbert's tenth problem by Matiyasevich. Hilbert asked for a procedure to determine whether a given polynomial Diophantine equation is solvable. This was a number theoretic problem, and he did not expect that such procedure can not exist. Proving non-existence though required developing a branch of mathematical logic now called computability theory (by Gödel, Church, Turing and others) that formalizes the notion of algorithm. Matiyasevich showed that any recursively enumerable set can be the solution set for a Diophantine equation, and since not all recursively enumerable sets are computable there can be no solvability algorithm.
This example is typical of how logic serves all parts of mathematics by saving effort on doomed searches for impossible constructions, proofs or counterexamples. For instance, an analyst might ask if the plane can be decomposed into a union of two sets, one at most countable along every vertical line, and the other along every horizontal line. It seems unlikely and people could spend a lot of time trying to disprove it. In vain, because Sierpinski proved that existence of such a decomposition is equivalent to the continuum hypothesis, and Gödel showed that disproving it is impossible by an elaborate logical construction now called inner model. As is proving it as Cohen showed by an even more elaborate logical construction called forcing.
A more recent example is the proof of the Mordell-Lang conjecture for function fields by Hrushovski (1996). The conjecture "is essentially a finiteness statement on the intersection of a subvariety of a semi-Abelian variety with a subgroup of finite rank". In characteristic $0$, and for Abelian varieties or finitely generated groups the conjecture was proved more traditionally by Raynaud, Faltings and Vojta. They inferred the result for function fields from the one for number fields using a specialization argument, another proof was found by Buium. Abramovich and Voloch proved many cases in characteristic $p$. Hrushovski gave a uniform proof for general semi-Abelian varieties in arbitrary characteristic using "model-theoretic analysis of the kernel of Manin's homomorphism", which involves definable subsets, Morley dimension, $\lambda$-saturated structures, model-theoretic algebraic closure, compactness theorem for first-order theories, etc.
The nonexistence of Fermat primes beyond $65537$ has been verified, as far as I know, to $2^{2^{32}}+1$. Thereby, we have identified the last constructible regular prime-sided polygon up to a point far beyond where any such construction could be carried out. (Based on current theories of quantum gravity, a regular polygon having the shortest possible side length and "only" $2^{2^8}+1$ sides would not fit in the known Universe.)
It was, of course, Euler who first killed Fermat's conjecture that $2^{2^n}+1$ is prime for all natural numbers $n$, by disproving it for $n=5$. Now the opposite conjecture is in vogue, and it has been verified up to $n=32$. Testing of Fermat numbers for primality can be accomplished by Pepin's Test, a stronger form of Fermat's Little Theorem whereby a Fermat number $M\ge 5$ is prime iff $3^{(M-1)/2}\equiv -1 \bmod M$. Because Pepin's test does not directly identify factors when the number is composite, $2^{2^n}+1$ has no known factors, despite being certified composite, for $n=20$ and $n=24$.
See here for a more thorough discussion of Fermat numbers.
Best Answer
Theorem: There is a surjective group homomorphism from the group $SU(2)$ of all $2\times2$ complex unitary matrices with determinant $1$ onto the group $SO(3,\mathbb R)$ of all $3\times3$ orthogonal matrices with determinant $1$.
Sketch of proof: See $SU(2)$ as$$\left\{a+bi+cj+dk\in\mathbb H\,\middle|\,a^2+b^2+c^2+d^2=1\right\}.$$Let$$\operatorname{Im}\mathbb H=\left\{\alpha i+\beta j+\gamma k\,\middle|\,\alpha,\beta,\gamma\in\mathbb R\right\}.$$Then, for each $q\in SU(2)$ and each $r\in\operatorname{Im}\mathbb H$, $qrq^{-1}\in\operatorname{Im}\mathbb H$. It turns out that the linear map $r\mapsto qrq^{-1}$ has determinant $1$. Furthermore, it preserves the quaternionic norm. So, we can see this map as en element of $SO(3,\mathbb R)$ and this map from $SU(2)$ into $SO(3,\mathbb R)$ is actually surjective (and its kernel is $\pm\operatorname{Id}$).